Question

The ratio of the lengths of two pieces of ribbon is 1:3. If 4 ft were cut from each piece, the sum of the new lengths would be 4 ft. How long would each piece be?

Answer 1

One piece has length `3` feet, the other has length `9` feet.

Explanation:

If the ratio of the length of the two pieces is `1/3`, then if `a` is the length of the small piece, the big piece will have length `3a`. If we cut `4` feet from each piece, their lengths are now

`a - 4` and `3a - 4`.

So, we know that their new lengths' sum is `4` feet, or

`(a - 4) + (3a - 4) = 4 => 4a - 8 = 4 => 4a = 12 => a = 3`

So one piece would have length `3` feet, and the other, `9` feet.

However, this problem seems a little weird, since we can't really cut `4` feet from a piece of length `3` feet. Nonetheless, a first degree equation, without any involvement of absolute values, can only have one root, and since the root is `a = 3` and the length of the other piece depends directly on this value, there are no other possible solutions to the problem.